Question 12.11.3: Consider again the macroeconomic model of Example 12.10.1. I......
Consider again the macroeconomic model of Example 12.10.1. If we assume that f , h, and m are differentiable functions with 0 < f^{\prime}< 1, h^{\prime}< 0, \text{and} m^{\prime}< 0, then these equations will determine Y, C, I, and r as differentiable functions of M, T, and G.
(a) Differentiate the system and express the differentials of Y, C, I, and r in terms of the differentials of M, T, and G. Find ∂Y/∂T and ∂C/∂T, and comment on their signs.
(b) Suppose moreover that P_{0} = (M_{0}, T_{0},G_{0}, Y_{0},C_{0}, I_{0}, r_{0}) is an initial equilibrium point for the system. If the money supply M, tax revenue T, and public expenditure G are all slightly changed as a result of government policy or central bank intervention, find the approximate changes in national income Y and in consumption C.
Taking differentials of Eqs (i)-(iv) in Example 12.1.1 yields
\mathrm{d}Y=\mathrm{d}C+\mathrm{d}I+\mathrm{d}G (v)
\mathrm{d}C=f^{\prime}(Y-T)(\mathrm{d}Y-\mathrm{d}T) (vi)
\mathrm{d}I=h^{\prime}(r)\,\mathrm{d}r (vii)
\mathrm{d}r=m^{\prime}(M)\,\mathrm{d}M (viii)
We wish to solve this linear system for the differential changes dy, dC, dI, and dr in the endogenous variables Y, C, I, and r, expressing these differentials in terms of the differentials of the exogenous policy variables dM, dT, and dG. From Eqs (vii) and (viii), we can find dI and dr immediately: dr = m^{\prime}(M) dM and dI = h^{\prime}(r)m^{\prime}(M) dM. Inserting the expression for dI into (v), while also rearranging (vi), we obtain the system:
\mathrm{d}Y-\mathrm{d}C=h^{\prime}(r)m^{\prime}(M)\,\mathrm{d}M+\mathrm{d}G;\quad f^{\prime}(Y-T)\,\mathrm{d}Y-\mathrm{d}C=f^{\prime}(Y-T)\,\mathrm{d}TThese are two equations to determine the two unknowns dY and dC. Solving for dY and dC, using a simplified notation, we get
\mathrm{d}Y={\frac{h^{\prime}m^{\prime}}{1-f^{\prime}}}\,\mathrm{d}M-{\frac{f^{\prime}}{1-f^{\prime}}}\,\mathrm{d}T+{\frac{1}{1-f^{\prime}}}\,\mathrm{d}G (ix)
\mathrm{d}C={\frac{f^{\prime}h^{\prime}m^{\prime}}{1-f^{\prime}}}\,\mathrm{d}M-{\frac{f^{\prime}}{1-f^{\prime}}}\,\mathrm{d}T+{\frac{f^{\prime}}{1-f^{\prime}}}\,\mathrm{d}G (x)
which express the differentials dY and dC as linear functions of the differentials dM, dT,and dG. Moreover, the solution is valid because f^{\prime}< 1 by assumption.
From the four equations (vii)–(x), it is easy to find the partial derivatives of Y, C, I, and r w.r.t. M, T, and G. For example, ∂Y/∂T = ∂C/∂T = −f ^{\prime}/(1 − f ^{\prime}) and ∂r/∂T = 0. Note that because 0 < f^{\prime}< 1, we have ∂Y/∂T = ∂C/∂T < 0. Thus, a small increase in the tax level, keeping M and G constant, decreases GDP, unless the extra tax revenue is all spent by the government. For if dT = dG = dx (and dM = 0), then dY = dx and dC = dI = dr =0.
If dM, dT, and dG are small in absolute value, then
\Delta Y=Y(M_{0}+\mathrm{d}M,T_{0}+\mathrm{d}T,G_{0}+\mathrm{d}G)-Y(M_{0},T_{0},G_{0})\approx\mathrm{d}YWhen computing dY, the partial derivatives are evaluated at the equilibrium point P_{0}.